The Lax pairs for the Holt system

By using non-canonical transformation between the Holt system and the Henon-Heiles system the Lax pairs for all the integrable cases of the Holt system are constructed from the known Lax representations for the Henon-Heiles system.Comment: 7 pages, LaTeX2e, a4.st

Duality between integrable Stackel systems

For the Stackel family of the integrable systems a non-canonical transformation of the time variable is considered. This transformation may be associated to the ambiguity of the Abel map on the corresponding hyperelliptic curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax representations and the dynamical r-matrix algebras are constructed. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail.Comment: LaTeX2e, 18 page

On the Drach superintegrable systems

Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated St\"ackel problems with quadratic integrals of motion. For the superintegrable St\"ackel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.Comment: 16 pages, Latex2e+Amssym

Higher-order superintegrability of a Holt related potential

In a recent paper, Post and Winternitz studied the properties of two-dimensional Euclidean potentials that are linear in one of the two Cartesian variables. In particular, they proved the existence of a potential endowed with an integral of third-order and an integral of fourth-order. In this paper we show that these results can be obtained in a more simple and direct way by noting that this potential is directly related with the Holt potential. It is proved that the existence of a potential with higher order superintegrability is a direct consequence of the integrability of the family of Holt type potentials.Comment: to appear in J. of Phys. A (2013

The Maupertuis principle and canonical transformations of the extended phase space

We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Various parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles variables. In this review we will discuss namely these induced transformations instead of the various parametric form of the geometric objects

A Refutation of Bell's Theorem

Bell's Theorem was developed on the basis of considerations involving a linear combination of spin correlation functions, each of which has a distinct pair of arguments. The simultaneous presence of these different pairs of arguments in the same equation can be understood in two radically different ways: either as `strongly objective,' that is, all correlation functions pertain to the same set of particle pairs, or as `weakly objective,' that is, each correlation function pertains to a different set of particle pairs. It is demonstrated that once this meaning is determined, no discrepancy appears between local realistic theories and quantum mechanics: the discrepancy in Bell's Theorem is due only to a meaningless comparison between a local realistic inequality written within the strongly objective interpretation (thus relevant to a single set of particle pairs) and a quantum mechanical prediction derived from a weakly objective interpretation (thus relevant to several different sets of particle pairs).Comment: RevTex4, 9 pages. Extended and entirely revised version. A talk given at the Vaxjo conference, Sweden; Nov. 2000. Submited to J. Math. Phy

Diversification and expression of the PIN, AUX/LAX, and ABCB families of putative auxin transporters in \u3cem\u3ePopulus\u3c/em\u3e

Intercellular transport of the plant hormone auxin is mediated by three families of membrane-bound protein carriers, with the PIN and ABCB families coding primarily for efflux proteins and the AUX/LAX family coding for influx proteins. In the last decade our understanding of gene and protein function for these transporters in Arabidopsis has expanded rapidly but very little is known about their role in woody plant development. Here we present a comprehensive account of all three families in the model woody species Populus, including chromosome distribution, protein structure, quantitative gene expression, and evolutionary relationships. The PIN and AUX/LAX gene families in Populus comprise 16 and 8 members respectively and show evidence for the retention of paralogs following a relatively recent whole genome duplication. There is also differential expression across tissues within many gene pairs. The ABCB family is previously undescribed in Populus and includes 20 members, showing a much deeper evolutionary history, including both tandem and whole genome duplication as well as probable gene loss. A striking number of these transporters are expressed in developing Populus stems and we suggest that evolutionary and structural relationships with known auxin transporters in Arabidopsis can point toward candidate genes for further study in Populus. This is especially important for the ABCBs, which is a large family and includes members in Arabidopsis that are able to transport other substrates in addition to auxin. Protein modeling, sequence alignment and expression data all point to ABCB1.1 as a likely auxin transport protein in Populus. Given that basipetal auxin flow through the cambial zone shapes the development of woody stems, it is important that we identify the full complement of genes involved in this process. This work should lay the foundation for studies targeting specific proteins for functional characterization and in situ localization

Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems

We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map completely integrable system into other completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and R-matrices. As an example, we consider canonical transformations of the extended phase space for the Toda lattices and the Stackel systems.Comment: LaTeX2e + Amssymb, 22p